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Nonlinear evolution of small disturbances into roll waves in an inclined open channel

Published online by Cambridge University Press:  26 April 2006

J. Yu
Affiliation:
Department of Mathematics and Statistics, University of Vermont. Burlington. VT 05405,USA
J. Kevorkian
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle. WA 98195. USA

Abstract

This paper concerns the asymptotic behaviour (for the limiting case of small amplitudes) of small disturbances as they evolve in time to produce the quasi-steady pattern of roll waves first discussed by Dressler in 1949. Roll waves exist if F, the undisturbed Froude number (dimensionless speed) of the flow, exceeds 2, and consist of a periodic pattern of bores separating two special continuous solutions of the governing equations in a uniformly translating frame. The mathematical problem is rather interesting as solutions of the linearized equations are unstable for F > 2. Thus, it is crucial to account for the cumulative effect of small nonlinearities to obtain a correct description of the flow over long times. We concentrate on the weakly unstable problem (0 < F − 2 [Lt ] 1) and use multiple scale expansions to derive the dominant evolution equation that governs the solution behaviour for long times. This turns out to be an integro-partial differential equation of first order that we solve numerically in conjunction with the jump condition that follows from the exact bore conditions. We present asymptotic and numerical results for periodic as well as isolated initial disturbances, and show that our theory predicts the solution accurately for both the transient and quasi-steady phases.

Type
Research Article
Copyright
© 1992 Cambridge University Press

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References

Chang, H. C. 1986 Phys. Fluids 29, 31423147.
Dressler, R. F. 1949 Commun. Pure Appl. Maths 2, 149194.
Hwang, S. H. & Chang, H. C. 1987 Phys. Fluids 30, 12591268.
Kawahara, T. & Toh, S. 1985 Phys. Fluids 28, 16361638.
Kevorkian, J. 1990 Partial Differential Equations: Analytical Solution Techniques. Wadsworth & Brooks/Cole Advanced Books & Software.
Kevorkian, J. & Yu, J. 1989 J. Fluid Mech. 204, 3156.
Nakaya, C. 1975 Phys. Fluids 18, 14071412.
Needham, D. J. & Merkin, J. H. 1984 Proc. R. Soc. Lond. A 394, 259278
Stoker, J. J. 1957 Water Waves. Interscience.
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
Yu, J. 1988 Passage through the critical Froude number for shallow water waves over a variable bottom. Ph.D. thesis, University of Washington, Seattle, WA.
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